2 Answers
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There is no direct conversion for what you are asking. Curies are a measure of activity equal to $3.7 \cdot 10^7$ decays per second. Rad are a measure of absorbed dose equal to 1 Joule per kilogram. In order to determine how much radiation is received, you need to know things like the amount of time of exposure, distance between source and target, sheilding if applicable (and intervening material regardless), solid angle of target, and mass of the target.

I assume that you meant to write $^{131}I$ and I expect that you are implying that the source was ingested and so all of the radiation is absorbed. $^{131}I$ always decays by $\beta^-$ emission and about 90% of these decays have an average energy of 191 keV, 7% have an average energy of 96 keV, and 2% have an average energy of 69 keV. Thus, on average, the $\beta^-$ have an energy of about 180 keV which is about $2.9 \cdot 10^{-14} J$.

A 50 mCi source gives off $1.85\cdot 10^9 \beta^-$ particles per second so, in this case, we are looking at about 0.003 Joules per minute of exposure. If you divide this by the weight of the patient, it will tell you the absorbed dose in rad.

Now, $^{131}I$ has a biological half-life of about 100 days. Biological half-life works about the same as radioactive half-life, there is an exponential decay of the effects. In other words, about 50 days after ingestion, you will be recieving only half of the exposure you would receive if the sample were, say, sitting on a table. Keep in mind, however, that the radioactive half-life of $^{131}I$ is only 8 days. You can combine radioactive half-life and biological half-life to determine the expected absorbed dose over time but I will not do it here.

One other thing to consider, however, is that the exposure to radiation in this case is not uniform because Iodine migrates to specific places in the body so that the residence time depends on which organ you are considering. Additionally, for health purposes we typically use dose equivalent rather than absorbed dose which is measured in Sieverts rather than rads. Dose equivalent is determined by multiplying the absorbed dose by a quality factor Q as recommended by the International Commission on Radiological Protection (ICRP). This accounts for the difference resulting from the energy of the incident particles. To make matters even more confusing, there is often also an "equivalent dose" (not the same as dose equivalent) that includes an additional weighting factor to account for the different nature of radiation from varous kinds of particles. On top of that, there is an "effective dose" that is different still and includes an additional weighting factor to account for the fact that a dose to the gonads is worse than one to bone.

In conclussion, yes, the suggested result you heard of 35 rad as an absorbed dose received from $^{131}I$ sounds reasonable. However, simplifying radiation exposure is frought with difficulty, every exposure situation is unique and needs individual and competant medical attention. Many of us in the industry wish the situation were simpler because the current situation is a public relations nightmare when something like Fukushima happens. Unfortunately, when you are dealing with the effect a stochastic process on biological systems, it just isn't that simple.

Thank you so much. I am asking as a patient (thyroid) and am trying to determine the magnitude of the ingested dose I received. One of the personnel told me that I probably absorbed the same amount of radiation as you would normally receive in about a year. This seems like it must be incorrect, and that it must be much higher.
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S. DanielsonAug 29 '12 at 18:07

I'm not sure I would say it is "much" higher. 35 rad would contribute on the order of magnitude of 300 mSv which is certainly quite high but that takes a lot of assumptions into account. If you are very concerned, do a bit of research; I work in the nuke power industry and am not a health physicist at that so my answer is more hueristic than quantitative result.
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AdamRedwineAug 29 '12 at 18:20

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Are you sure, for example, that the activity was milicuries and not microcuries?
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AdamRedwineAug 29 '12 at 18:22

That's interesting. I have tried to look up the information but as there is no direct conversion as you said, its difficult for a layman to get an idea of magnitude compared to things we have a sense of already, like X rays. I'm guessing you answered as you did because the dose to the thyroid in RAI therapy is much higher than to the rest of the body, even though that would be still higher than what would be normally absorbed in a year? I wondered about this too because there are so many precautions advised for the people around you when you have the dose, yet they insist that the treatment is
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S. DanielsonAug 29 '12 at 19:00

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@S.Danielson Hi, and welcome to Physics Stack Exchange! As dmckee told you earlier, please do not respond to other people's answers by posting another answer of your own. Use the "add comment" link which appears just below this message at the left.
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David Z♦Aug 29 '12 at 19:54

AdamRedwine's answer also doesn't take into account the fact that the radioactive iodine will be processed by the body and expelled in the urine over the course of a few days. This is what the comment on the increased exposure on the bladder wall is from, referenced by another comment. So I believe (but am not an expert in this area) that the exposure will actually be much less than just taking the (already short) half-life into account.

[Apologies, I want to provide a comment on the previous answer, but it seems like the well-meaning but misguided moderation system won't let me.]